I believe that mathematics should be taught, not collaboratively explored; algebra and geometry are better than a vague course of Integrated Math; spiraling doesn't work nearly as well as learning it properly the first time; "I don't DO math" should be an incentive rather than an excuse. "I don't DO English" should be treated the same way.

The criticism of traditional math teaching is based largely on a
mischaracterization of how it is/has been taught, and misrepresented as
having failed thousands of students in math education despite evidence
of its effectiveness in the 1940’s, 50’s and 60’s.

Reacting to this
characterization of the traditional model, math reformers promote a
teaching approach in which understanding and process dominate over
content. In lower grades, mental math and number sense are emphasized
before students are fluent with procedures and number facts.

Procedural
fluency is seldom achieved.

In lieu of the standard methods for
adding/subtracting, multiplying and dividing, in some programs students
are taught strategies and alternative methods. Whole class and
teacher-led explicit instruction (and even teacher-led discovery) has
given way to what the education establishment believes is superior:
students working in groups in a collaborative learning environment.
Classrooms have become student-centered and inquiry-based. The grouping
of students by ability has almost entirely disappeared in the lower
grades—full inclusion has become the norm.

Reformers dismiss the
possibility that understanding and discovery can be achieved by students
working on sets of math problems individually and that procedural
fluency is a prerequisite to understanding.

Much of the education
establishment now believes it is the other way around; if students have
the understanding, then the need to work many problems (which they term
“drill and kill”) can be avoided.
The de-emphasis on mastery of basic facts, skills and procedures has
met with growing opposition, not only from parents but also from
university mathematicians. At a recent conference on math education held
in Winnipeg, math professor Stephen Wilson from Johns Hopkins
University said, much to the consternation of the educationists on the
panel, that “the way mathematicians learn is to learn how to do it first
and then figure out how it works later.” This sentiment was also echoed
in an article written by Keith Devlin (2006). Such opposition has had
limited success, however, in turning the tide away from reform
approaches.

Here's the whole article:

Mathematics Education: Being Outwitted by Stupidity

By Barry Garelick
In a well-publicized paper that addressed why some students were not
learning to read, Reid Lyon (2001) concluded that children from
disadvantaged backgrounds where early childhood education was not
available failed to read because they did not receive effective
instruction in the early grades. Many of these children then required
special education services to make up for this early failure in reading
instruction, which were by and large instruction in phonics as the means
of decoding. Some of these students had no specific learning disability
other than lack of access to effective instruction. These findings are
significant because a similar dynamic is at play in math education: the
effective treatment for many students who would otherwise be labeled
learning disabled is also the effective preventative measure.
In 2010 approximately 2.4 million students were identified with
learning disabilities — about three times as many as were identified in
1976-1977. (See http://nces.ed.gov/programs/digest/d10/tables/xls/tabn045.xls and http://www.ideadata.org/arc_toc12.asp#partbEX).
This increase raises the question of whether the shift in instructional
emphasis over the past several decades has increased the number of low
achieving children because of poor or ineffective instruction who would
have swum with the rest of the pack when traditional math teaching
prevailed. I believe that what is offered as treatment for math learning
disabilities is what we could have done—and need to be doing—in the
first place. While there has been a good amount of research and effort
into early interventions in reading and decoding instruction, extremely
little research of equivalent quality on the learning of mathematics
exists. Given the education establishment’s resistance to the idea that
traditional math teaching methods are effective, this research is very
much needed to draw such a definitive conclusion about the effect of
instruction on the diagnosis of learning disabilities.1Some Background
Over the past several decades, math education in the United States
has shifted from the traditional model of math instruction to “reform
math”. The traditional model has been criticized for relying on rote
memorization rather than conceptual understanding. Calling the
traditional approach “skills based”, math reformers deride it and claim
that it teaches students only how to follow the teacher’s direction in
solving routine problems, but does not teach students how to think
critically or to solve non-routine problems. Traditional/skills-based
teaching, the argument goes, doesn’t meet the demands of our 21st
century world.
As I’ve discussed elsewhere,
the criticism of traditional math teaching is based largely on a
mischaracterization of how it is/has been taught, and misrepresented as
having failed thousands of students in math education despite evidence
of its effectiveness in the 1940’s, 50’s and 60’s. Reacting to this
characterization of the traditional model, math reformers promote a
teaching approach in which understanding and process dominate over
content. In lower grades, mental math and number sense are emphasized
before students are fluent with procedures and number facts. Procedural
fluency is seldom achieved. In lieu of the standard methods for
adding/subtracting, multiplying and dividing, in some programs students
are taught strategies and alternative methods. Whole class and
teacher-led explicit instruction (and even teacher-led discovery) has
given way to what the education establishment believes is superior:
students working in groups in a collaborative learning environment.
Classrooms have become student-centered and inquiry-based. The grouping
of students by ability has almost entirely disappeared in the lower
grades—full inclusion has become the norm. Reformers dismiss the
possibility that understanding and discovery can be achieved by students
working on sets of math problems individually and that procedural
fluency is a prerequisite to understanding. Much of the education
establishment now believes it is the other way around; if students have
the understanding, then the need to work many problems (which they term
“drill and kill”) can be avoided.
The de-emphasis on mastery of basic facts, skills and procedures has
met with growing opposition, not only from parents but also from
university mathematicians. At a recent conference on math education held
in Winnipeg, math professor Stephen Wilson from Johns Hopkins
University said, much to the consternation of the educationists on the
panel, that “the way mathematicians learn is to learn how to do it first
and then figure out how it works later.” This sentiment was also echoed
in an article written by Keith Devlin (2006). Such opposition has had
limited success, however, in turning the tide away from reform
approaches.The Growth of Learning Disabilities
Students struggling in math may not have an actual learning
disability but may be in the category termed “low achieving” (LA).
Recent studies have begun to distinguish between students who are LA and
those who have mathematical learning disabilities (MLD). Geary (2004)
states that LA students don’t have any serious cognitive deficits that
would prevent them from learning math with appropriate instruction.
Students with MLD, however, (about 5-6% of students) do appear to have
both general (working memory) and specific (fact retrieval) deficits
that result in a real learning disability. Among other reasons,
ineffective instruction, may account for the subset of LA students
struggling in mathematics.
The Individuals with Disabilities Education Act (IDEA) initially
established the criteria by which students are designated as “learning
disabled”. IDEA was reauthorized in 2004 and renamed the Individuals
with Disabilities Education Improvement Act (IDEIA). The reauthorized
act changed the criteria by which learning disabilities are defined and
removed the requirements of the “significant discrepancy” formula. That
formula identified students as learning disabled if they performed
significantly worse in school than indicated by their cognitive
potential as measured by IQ. IDEIA required instead that states must
permit districts to adopt alternative models including the “Response to
Intervention” (RtI) model in which struggling students are pulled out of
class and given alternative instruction.
What type of alternative instruction is effective? A popular textbook
on special education (Rosenberg, et. al, 2008), notes that up to 50% of
students with learning disabilities have been shown to overcome their
learning difficulties when given explicit instruction. This idea is
echoed by others and has become the mainstay of RtI. What Works
Clearinghouse finds strong evidence that explicit instruction is an effective intervention,
stating: “Instruction during the intervention should be explicit and
systematic. This includes providing models of proficient problem
solving, verbalization of thought processes, guided practice, corrective
feedback, and frequent cumulative review”. Also, the final report of the President’s National Math Advisory Panel
states: “Explicit instruction with students who have mathematical
difficulties has shown consistently positive effects on performance with
word problems and computation. Results are consistent for students with
learning disabilities, as well as other students who perform in the
lowest third of a typical class.” (p. xxiii). The treatment for low
achieving, learning disabled and otherwise struggling students in math
thus includes some of the traditional methods for teaching math that
have been decried by reformers as having failed millions of students.The Stealth Growth of Effective Instruction
Although the number of students classified as learning disabled has
grown since 1976, the number of students classified as LD since the
passage of IDEIA has decreased (see Figure 1). Why the decrease has
occurred is not clear. A number of factors may be at play. One may be a
provision of No Child Left Behind that allows schools with low numbers
of special-education students to avoid reporting the academic progress
of those students. Other factors include more charter schools, expanded
access to preschools, improved technologies, and greater understanding
of which students need specialized services. Last but not least, the
decrease may also be due to targeted RtI programs that have reduced the
identification of struggling and/or low achieving students as learning
disabled. .
Having seen the results of ineffective math curricula and pedagogy as
well as having worked with the casualties of such educational
experiments, I have no difficulty assuming that RtI plays a significant
role in reducing the identification of students with learning
disabilities. In my opinion it is only a matter of time before
high-quality research and the best professional judgment and experience
of accomplished classroom teachers verify it. Such research should
include 1) the effect of collaborative/group work compared to individual
work, including the effect of grouping on students who may have
difficulty socially; 2) the degree to which students on the autistic
spectrum (as well as those with other learning disabilities) may depend
on direct, structured, systematic instruction; 3) the effect of explicit
and systematic instruction of procedures, skills and problem solving,
compared with inquiry-based approaches; 4) the effect of sequential and
logical presentation of topics that require mastery of specific skills,
compared with a spiral approaches to topics that do not lead to closure
and 5) Identifying which conditions result in
student-led/teacher-facilitated discovery, inquiry-based, and
problem-based learning having a positive effect, compared with
teacher-led discovery, inquiry-based and problem-based learning. Would
such research show that the use of RtI is higher in schools that rely on
programs that are low on skills and content but high on trendy unproven
techniques and which promise to build critical thinking and higher
order thinking skills? If so, shouldn’t we be doing more of the RtI
style of teaching in the first place instead of waiting to heal reform
math’s casualties?
Until any such research is in, the educational establishment will
continue to resist recognizing the merits of traditional math teaching.
One education professor with whom I spoke stated that the RtI model fits
mathematics for the 1960s, when “skills throughout the K-8 spectrum
were the main focus of instruction and is seriously out of date.”
Another reformer argued that reform curricula require a good deal of
conceptual understanding and that students have to do more than solve
word problems. These confident statements assume that traditional
methods—and the methods used in RtI—do not provide this understanding.
In their view, students who respond to more explicit instruction
constitute a group who may simply learn better on a superficial level.
Based on these views, I fear that RtI will incorporate the pedagogical
features of reform math that has resulted in the use of RtI in the first
place.
While the criticism of traditional methods may have merit for those
occasions when it has been taught poorly, the fact that traditional math
has been taught badly doesn’t mean we should give up on teaching it
properly. Without sufficient skills, critical thinking doesn’t amount to
much more than a sound bite. If in fact there is an increasing trend
toward effective math instruction, it will have to be stealth enough to
fly underneath the radar of the dominant edu-reformers. Unless and until
this happens, the thoughtworld of the well-intentioned educational
establishment will prevail. Parents and professionals who benefitted
from traditional teaching techniques and environments will remain on the
outside — and the public will continue to be outwitted by stupidity.Source: U.S. Department of Education, National Center for
Education Statistics (2011). Digest of Education Statistics, 2010 (NCES
2011-015), Chapter 2.Barry Garelick has written extensively about
math education in various publications including Education Next,
Educational Leadership, and Education News. He recently retired from the
federal government and has completed his requirements for a credential
to teach math (middle school/high school) in California.1This article focuses on math teaching and learning, but
the same pedagogical issues arise in history, science, and English
Language Arts (ELA), including grammar, spelling, composition, reading
comprehension and literature.References
Devlin, Keith. (2006). Math back in forefront, but debate lingers on how to teach it. San Jose Mercury News. Feb. 19.
Geary, David. (2004). Mathematics and learning disabilities. J Learn Disabil 2004; 37; 4
Lyon, Reid (2001), in “Rethinking special education for a new century” (Chapter 12) by Chester Finn, et al., Thomas B. Fordham Foundation; Progressive Policy Inst., Washington, DC.
Available via http://eric.ed.gov/PDFS/ED454636.pdf
Rosenberg, Michael S., Westling, D.L., McLeskey, J. 2008. Special Education for Today’s Teachers: An Introduction. Columbus: Pearson, Merrill Prentice Hall.

8 comments:

I went to school in the 50's and 60's (and 70's, 80's, 90's . . . what can I say?). There is no flipping way the education I am giving my students approaches the education I experienced and they won't approach the overall understanding I entered college with.

They won't practice by doing one more problem than they feel :necessary".

Thank you for linking to my article. You may be interested in a recent article about the pedagogical agenda in the Common Core Math Standards. http://www.educationnews.org/education-policy-and-politics/the-pedagogical-agenda-of-common-core-math-standards/

How much of the "learning disabilities" are just the fact that we are actually trying to educate these students now? The problem is of course that we are trying to give them a "College Prep" curriculum, but give it in such a watered down way that it does not prepare them for college...or anything else. Hands-on group discovery learning used to be called "Vocational", but that's now a bad word in education.

This "freeskier" is also a parent (8 and 10 yrs.), who comments as one so as to not allow my 'professional opinions' to get me in trouble with my supervisors... But to the point - as a parent, I recommend the website, "extra math" to you. It is being used by Clarendon Elementary to supplement my daughter's learning (she's way, way ahead of her peers).Wouldn't you know? Timed calculation tests... just like the 40's, 50's and 60's.Needless to say, my daughter (and son, for that matter) continue to pull away from their peers, mathematically.

This is all hogwash. You would all do well to remember that in these "good old days" of the 50s, 60s, and 70s, it was acceptable for some people to be "good at math" and most people to feel like abject failures. Take a survey of folks who attended school during these decades. My money is on 80% of them claiming that they are not "math people." Ask them what makes some people "math people" and others not. They will tell you, they were fast and they just got it. This is unacceptable. Speed with procedures is NOT mathematics and yet it is the measuring tool that many victims of public education use to size themselves up, mathematically. In the "good old days" that many of you are inventing, school was not about educating all kids. It was about sorting kids out, the "cans" from the "cannots" and this was acceptable. This is not what education is about anymore; it is about education for ALL. And this is why everything you are saying about returning to a model meant for sorting is complete and utter nonsense.

About Me

I'm a high-school math teacher completely frustrated with new math, reform math, fuzzy math, the color of math, talking about math, literacy across the curriculum and all those other things that get in the way of students actually learning math, not to mention the ever-present "You need to help raise our scores by taking one day a week to go over test-taking skills" and other administrative folderol.